Competition between carriers has partly driven the widespread availability of coverage. Several carriers compete on the basis of their coverage. A government-funded entity seems highly unlikely to provide optimal coverage, or to do so in a cost-efficient manner.

I'm not categorically against government-run solutions. But the cellular market isn't so obviously broken that a government-run system has any hope of being better. U.S. market concentration in cellular is less than for shipping, ecommerce, and soft drinks, and a lot less than for mobile operating systems, search, or social media. Verizon and AT&T's combined market share of about 68% is about the same as Apple and Samsung's combined share of the U.S. smart phone market.

with 4G finally getting fast enough (along with the low latency promises of 5G), i think we're almost at the point where what you propose might make sense.

before this, wireless tech/infrastructure just wasn't good enough, and needed continuing investment to improve. now that we're over the hump, a GSE can begin buying up the 4G infrastructure and leaving operations/service to the carriers.

since there are 4 national and some regional carriers, you could even divvy the duplicative infrastructure up between state and federal GSEs, much like state and federal highways. then you'd have some semblance of competition that would keep the infrastructure from just crumbling away from political/bureaucratic neglect.